Octave 3.8, jcobi/2

Average Accuracy: 63.2% → 97.7%

Time: 29.4s

Precision: binary64

Cost: 35268

\[\left(\alpha > -1 \land \beta > -1\right) \land i > 0\]

Math

\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

↓

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.9999995:\\ \;\;\;\;\frac{\frac{0}{\alpha} + \frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(\mathsf{fma}\left(\alpha + \beta, \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)}}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)\right)}^{3}\right)}^{0.3333333333333333}}{2}\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (/
  (+
   (/
    (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i)))
    (+ (+ (+ alpha beta) (* 2.0 i)) 2.0))
   1.0)
  2.0))

↓

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<=
        (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0))
        -0.9999995)
     (/ (+ (/ 0.0 alpha) (/ (+ (* i 4.0) (+ 2.0 (* beta 2.0))) alpha)) 2.0)
     (/
      (pow
       (pow
        (fma
         (+ alpha beta)
         (/
          (/ (- beta alpha) (+ (+ alpha beta) (fma 2.0 i 2.0)))
          (+ alpha (fma 2.0 i beta)))
         1.0)
        3.0)
       0.3333333333333333)
      2.0))))

C

double code(double alpha, double beta, double i) {
	return (((((alpha + beta) * (beta - alpha)) / ((alpha + beta) + (2.0 * i))) / (((alpha + beta) + (2.0 * i)) + 2.0)) + 1.0) / 2.0;
}

↓

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.9999995) {
		tmp = ((0.0 / alpha) + (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha)) / 2.0;
	} else {
		tmp = pow(pow(fma((alpha + beta), (((beta - alpha) / ((alpha + beta) + fma(2.0, i, 2.0))) / (alpha + fma(2.0, i, beta))), 1.0), 3.0), 0.3333333333333333) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / Float64(Float64(alpha + beta) + Float64(2.0 * i))) / Float64(Float64(Float64(alpha + beta) + Float64(2.0 * i)) + 2.0)) + 1.0) / 2.0)
end

↓

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0)) <= -0.9999995)
		tmp = Float64(Float64(Float64(0.0 / alpha) + Float64(Float64(Float64(i * 4.0) + Float64(2.0 + Float64(beta * 2.0))) / alpha)) / 2.0);
	else
		tmp = Float64(((fma(Float64(alpha + beta), Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + fma(2.0, i, 2.0))) / Float64(alpha + fma(2.0, i, beta))), 1.0) ^ 3.0) ^ 0.3333333333333333) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]

↓

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], -0.9999995], N[(N[(N[(0.0 / alpha), $MachinePrecision] + N[(N[(N[(i * 4.0), $MachinePrecision] + N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Power[N[Power[N[(N[(alpha + beta), $MachinePrecision] * N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.999999500000000041

   1. Initial program 2.7%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   2. Simplified 14.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\alpha + \beta, \frac{\beta - \alpha}{\left(\alpha + \mathsf{fma}\left(2, i, \beta\right)\right) \cdot \left(\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)\right)}, 1\right)}{2}} \]
      Proof

      [Start] 2.7	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   3. Taylor expanded in alpha around inf 77.9%

      \[\leadsto \frac{\color{blue}{\left(\frac{\beta \cdot \left(\beta - -1 \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)\right)}{{\alpha}^{2}} + \left(-1 \cdot \frac{\beta}{\alpha} + \left(\frac{\beta}{\alpha} + -1 \cdot \frac{\left(\beta - -1 \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)\right) \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right) + -1 \cdot \left(\left(\beta + 2 \cdot i\right) \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}{{\alpha}^{2}}\right)\right)\right) - -1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}}{2} \]

   4. Taylor expanded in alpha around inf 90.4%

      \[\leadsto \frac{\color{blue}{\frac{-1 \cdot \beta + \beta}{\alpha}} - -1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}{2} \]

   5. Simplified 90.4%

      \[\leadsto \frac{\color{blue}{\frac{0}{\alpha}} - -1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}{2} \]
      Proof

      [Start] 90.4	\[ \frac{\frac{-1 \cdot \beta + \beta}{\alpha} - -1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}{2} \]
      distribute-lft1-in [=>] 90.4	\[ \frac{\frac{\color{blue}{\left(-1 + 1\right) \cdot \beta}}{\alpha} - -1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}{2} \]
      metadata-eval [=>] 90.4	\[ \frac{\frac{\color{blue}{0} \cdot \beta}{\alpha} - -1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}{2} \]
      mul0-lft [=>] 90.4	\[ \frac{\frac{\color{blue}{0}}{\alpha} - -1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}{2} \]

   if -0.999999500000000041 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))

   1. Initial program 81.2%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   2. Simplified 86.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\alpha + \beta, \frac{\beta - \alpha}{\left(\alpha + \mathsf{fma}\left(2, i, \beta\right)\right) \cdot \left(\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)\right)}, 1\right)}{2}} \]
      Proof

      [Start] 81.2	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   3. Applied egg-rr 99.9%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(\beta + \alpha, \frac{\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, 2\right) + \left(\beta + \alpha\right)}}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)\right)}^{3}\right)}^{0.3333333333333333}}}{2} \]
      Proof

      [Start] 86.8	\[ \frac{\mathsf{fma}\left(\alpha + \beta, \frac{\beta - \alpha}{\left(\alpha + \mathsf{fma}\left(2, i, \beta\right)\right) \cdot \left(\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)\right)}, 1\right)}{2} \]
      add-cbrt-cube [=>] 86.8	\[ \frac{\color{blue}{\sqrt[3]{\left(\mathsf{fma}\left(\alpha + \beta, \frac{\beta - \alpha}{\left(\alpha + \mathsf{fma}\left(2, i, \beta\right)\right) \cdot \left(\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)\right)}, 1\right) \cdot \mathsf{fma}\left(\alpha + \beta, \frac{\beta - \alpha}{\left(\alpha + \mathsf{fma}\left(2, i, \beta\right)\right) \cdot \left(\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)\right)}, 1\right)\right) \cdot \mathsf{fma}\left(\alpha + \beta, \frac{\beta - \alpha}{\left(\alpha + \mathsf{fma}\left(2, i, \beta\right)\right) \cdot \left(\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)\right)}, 1\right)}}}{2} \]
      pow1/3 [=>] 86.8	\[ \frac{\color{blue}{{\left(\left(\mathsf{fma}\left(\alpha + \beta, \frac{\beta - \alpha}{\left(\alpha + \mathsf{fma}\left(2, i, \beta\right)\right) \cdot \left(\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)\right)}, 1\right) \cdot \mathsf{fma}\left(\alpha + \beta, \frac{\beta - \alpha}{\left(\alpha + \mathsf{fma}\left(2, i, \beta\right)\right) \cdot \left(\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)\right)}, 1\right)\right) \cdot \mathsf{fma}\left(\alpha + \beta, \frac{\beta - \alpha}{\left(\alpha + \mathsf{fma}\left(2, i, \beta\right)\right) \cdot \left(\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)\right)}, 1\right)\right)}^{0.3333333333333333}}}{2} \]

3. Recombined 2 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.9999995:\\ \;\;\;\;\frac{\frac{0}{\alpha} + \frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(\mathsf{fma}\left(\alpha + \beta, \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)}}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)\right)}^{3}\right)}^{0.3333333333333333}}{2}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.7%
Cost	22340

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.9999995:\\ \;\;\;\;\frac{\frac{0}{\alpha} + \frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)}, \frac{\alpha + \beta}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2}\\ \end{array} \]

Alternative 2
Accuracy	97.7%
Cost	16068

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.9999995:\\ \;\;\;\;\frac{\frac{0}{\alpha} + \frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{2}\\ \end{array} \]

Alternative 3
Accuracy	95.3%
Cost	5192

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0}\\ \mathbf{if}\;t_1 \leq -0.9999995:\\ \;\;\;\;\frac{\frac{0}{\alpha} + \frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-52}:\\ \;\;\;\;\frac{t_1 + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}{2}\\ \end{array} \]

Alternative 4
Accuracy	82.7%
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 4 \cdot 10^{+18}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{\alpha} + \frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 5
Accuracy	78.2%
Cost	973

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 6.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 4.4 \cdot 10^{+220} \lor \neg \left(\alpha \leq 7.6 \cdot 10^{+277}\right):\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 6
Accuracy	74.7%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 2.8 \cdot 10^{+122}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + 2}{\alpha}}{2}\\ \end{array} \]

Alternative 7
Accuracy	77.3%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 2.05 \cdot 10^{+86}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 8
Accuracy	79.4%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 10^{+40}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 9
Accuracy	71.8%
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5.8 \cdot 10^{+91}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10
Accuracy	62.1%
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023151 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (and (> alpha -1.0) (> beta -1.0)) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0)) 1.0) 2.0))